What you have so far is perfect. You started with the given statements and then use the definition of midpoint to show that each piece is equal to one another.
From here, use the transitive property
We combine these two statements
into the following
WX = YZ
So we can effectively say WX = XY = YZ meaning the segments WX, XY and YZ are all the same length.
Through similar logic, we can chain together
to get VW = XY
-------------
Consequently this means VW, WX, XY, and YZ are all the same length
Let's call that length m.
m = length of segment VW = length of WX = etc etc
This means,
VY = VW+WX+XY = m+m+m = 3m
and
WZ = WX+XY+YZ = m+m+m = 3m
Both VY and WZ are exactly 3m units long, therefore, VY = WZ
